On the choice of variance for the log rank test

SUMMARY The log rank test is widely used for comparison of survival curves. This paper examines various estimators for the variance of the log rank test statistic. These include the Mantel-Haenszel variance, the permutation variance of Peto & Peto and several newly proposed estimators. The results generalize to a class of test statistics proposed by Tarone & Ware. The log rank test is widely used for comparison of survival curves. There are two versions of the test which differ in the choice of variance for the same test statistic T. These are the Mantel-Haenszel variance, an estimate of var (T) which is unbiased independently of differences in censoring, and the permutation variance of Peto & Peto (1972), for which the variance is computed under the assumption of equal censoring. This paper examines the choice of variance. It is observed that for equal sample sizes the Mantel-Haenszel variance tends to underestimate the true variance when T is large in absolute value. As this is the situation where rejection of the null hypothesis is to be seriously considered, it appears that the true type I error will tend to be higher than that assumed, and the reported P value will tend to be too low. This phenomena is discussed also by Morton (1978). In ? 4, it is shown that under equal sample sizes the permutation variance tends to overestimate the true variance when censoring is unequal. It thus appears prudent to use the permutation variance rather than the Mantel-Haenszel variance in assessing significance. In ? 5 alternative variance estimators are proposed. The results immediately generalize to a class of test statistics proposed by Tarone & Ware (1977). This point is discussed in ? 6.